/
Text
An intrinsic tensor technique in Minkowski space
with applications to boundary value problems
George N. Borzdov
Department of Physics, Byelorussian State University, Minsk 220050, Belarus
(Received 7 January 1993; accepted for publication 17 February 1993)
This article describes an intrinsic tensor technique for solving boundary value
problems in electrodynamics of moving media, which entails considerable nota-
tional simplification and calculational advantages. On the basis of generalized
impedance and characteristic matrix methods, the reflection and transmission
operators of a multilayer structure consisting of anisotropic media uniformly
moving with different velocities are found. A general solution to the problem of
reflection and transmission at the interface between two moving isotropic media
is obtained, and explicit expressions for Doppler shifts, Snell’s laws, critical and
Brewster angles, reflection and transmission coefficients, and the radiation pres-
sure force are found. The material tensor of a uniformly moving linear medium
is expressed through its reflection and transmission operators.
I. INTRODUCTION
It is well-known1"6 that the intrinsic representation of vectors, tensors, and spinors can be
very useful in solving various problems in electrodynamics and special relativity. Calculational
advantages of intrinsic tensor techniques based on the use of exterior differential forms were
discussed in detail by Misner, Thorne, and Wheeler,1 Luehr and Rosenbaum,4 and De-
schamps.5 These techniques can be a particularly useful supplement to the vector, dyadic, and
matrix methods which are now widely used in electrodynamics of moving media. Many prob-
lems of relativistic electrodynamics arouse considerable current interest of mathematical phys-
icists, as well as applied physicists, electrical engineers, and experimentalists. Within the last
three decades such subjects as constitutive relations7"11 for uniformly moving linear media,
radiation and propagation9"17 of electromagnetic waves in such media, problems of reflection
and transmission of electromagnetic waves by uniformly moving semi-infinite dielectric1*”32
and plasma33”37 media, dielectric38 and plasma39 slabs, some types of anisotropic media10,34,35,40
as well as moving interfaces between stationary isotropic media,11,21,26,27 and some types of
moving multilayer systems21,41 have been extensively discussed in the literature. Various inter-
esting features concerning Doppler shifts,10,1 Ы6*|8-21»26*27»35,38 gCnerapzcd Snell’s
laws,11’18”21’26”28’35 critical11,19,24,29,34,35 and Brewster11119,20,25,26,30,31,34-36 angles, reflection and
transmission coefficients,10,11’18-27,33-35,38”41 and pressure of light on a moving dielectric19,22,32
have been uncovered.
To solve boundary value problems in electrodynamics of uniformly moving media, two
methods are mainly used: (a) the method based upon the principle of phase invariance, the
covariance of Maxwell’s equations and the Lorentz transformations for field vectors and wave
four-vectors (Yeh,18,19,33,38 Shiozawa, Hazama and Kamagai,23,24 Kritikos, Lee, and Papas,41
Chawla and Unz,39 Kalluri and Prasad33); (b) the method based upon Maxwell’s equations
and the constitutive relations for a moving medium (Tai,7 Pyati,20 Daly and Grucnbcrg,*'2 Lee
and Lo,34 Cheng and Kong,9,10,35 Bolotovskii and Stolyrov,26,27 Saca28). Both methods have
their merits and demerits. In method (a) the fields arc first transformed to the proper frame K'
of the moving medium, in K' the problem is solved as for a stationary medium, and the
obtained field expressions are then transformed back to the laboratory frame Л. This method
becomes inconvenient to apply when there does not exist a common rest frame for all material
0022-2488/93/34(7)/3162/35/$6.00
(c) 1993 American Institute of Physics
media and interfaces. In method (b), on the basis of Minkowski constitutive relations42 and, in
the general case, the concept of a bianisotropic medium,8-10,35 the problem is solved directly in
the frame ЛЛ However, the dependence of material parameters of bianisotropic medium in К on
its velocity and material parameters in K' is usually quite involved.
On the other hand, numerous useful methods of solving boundary value problems, such as
the methods of Green’s functions, geometric asymptotics, Jones’ matrices, impedances and
characteristic matrices (evolution operators), are developed in the theory of wave propagation
in inhomogeneous stationary media, including isotropic and anisotropic layered media.43-56 In
the theory of layered media, the most general and effective methods are based on the use of the
impedance concept and the characteristic matrices introduced by Abeles.50 However, the im-
pedance and characteristic matrix methods are formulated in a noncovariant form and that is
why they still do not take a proper place in relativistic electrodynamics. Meanwhile, these
methods, formulated in Lorentz covariant form, can be very useful in investigating various
moving isotropic and anisotropic layered media and multiple moving interfaces in a stationary
medium, which have many interesting features, as well as important practical applica-
tions.21’26’27’41’57
The purpose of this article is threefold: (i) to present an intrinsic tensor technique in
Minkowski space, which entails considerable notational simplification and calculational advan-
tages; (ii) to generalize, using this technique, the impedance and characteristic matrix methods
on the case of uniformly moving linear media; and (iii) to illustrate the advantages implicit in
these methods by applying them to some boundary value problems. We do not apply here the
formalism based on the introduction of the imaginary fourth coordinate ict. It eliminates the
need to distinguish between contravariant and covariant components but has some substantial
drawbacks.1,58 In this article, we show that exactly the use of p-covariant 7-contravariant
tensors is a very natural and convenient approach to the development of coordinate-free tensor
techniques in Minkowski space. We share the point of view (for example, see Misner, Thorne,
and Wheeler ) that there is no necessity or sufficiently valuable gain to disguise the vital
distinctions between Minkowski space and Euclidean space or between 5-vectors and s-forms.
An interesting abstract formalism, based on an extension of the intrinsic concepts of vectors
and tensors to Minkowski space, was suggested earlier by Luehr and Rosenbaum.4 In the frame
of their approach, due to the introduction of some specific system of notation, manipulations
with vectors and tensors become formally the same as those in Euclidean space. We suggest
here another intrinsic tensor technique. It is based on the use of dual exterior algebras and
antisymmetric tensors which describe linear mappings in the spaces of г-vectors and s-forms.
This technique is more closely related with the standard system of notation of exterior algebra
and is more convenient for applications to boundary value problems.
The plan of the article is as follows. The intrinsic tensor technique in Minkowski space is
discussed in Sec. II. Basic Lorentz covariant equations and evolution operators for waves in a
moving multilayer structure, and properties of eigenwaves in a homogeneous medium are
considered in Secs. Ill, IV, and V, respectively. In Secs. VI and VII, the impedance and
characteristic matrix methods are generalized on the case of moving media and then applied to
find the reflection and transmission operators of the interface between two media and of a
multilayer structure. The general case of reflection and transmission of waves at the interface
between two moving isotropic media, a wave with cubic amplitude dependence on coordinates
in a biaxially anisotropic layer, and a method for generalization of obtained results on the case
of dispersive media are considered in Sec. VIII. In Sec. IX, explicit expressions are obtained,
which enable one to find all 36 components М^к1 of the material tensor of a moving medium,
given its reflection and transmission operators.
IL INTRINSIC TENSOR FORMALISM IN MINKOWSKI SPACE
A. Antisymmetric tensors
Let Land И* be the four-dimensional Minkowski vector space and its dual. Consider the
spaces (Refs. 59-62) Г/— P® ••• 0 И 0 •••«И and Trx— У ®
r-contravariant s-covariant tensors [tensors of type ()]. Let A' and Ars denote the subspaces
of Tsr and Trs, consisting of antisymmetric tensors. Elements of J/=Hr0=Ar( K) and A °
=A°s^As(y*) are called г-vcctors (antisymmetric r-contravariant tensors) and s-forms (an-
tisymmetric s-covariant tensors). The duality of the exterior (Grassman) algebras A(L) and
Л( И*) is defined by monomials v = Vj A • • • A vrG Ar( И) and = A • • • Ao/eA5( И*) as fol-
lows (Refs. 59-62): (v,ru) =dct( <v,,,*?)) if г=л, and (v,w)=O if г/л, where v,G И, wyeP,
~cu7 (vf) is the value of the linear form coJ at the vector v,, A is the exterior product.
In this article, we distinguish the r-vectors from other tensors by the bold type. As usual, to
number r-vcctors and 5-forms, wc use subscripts and superscripts, respectively.
Let g = 2;Li ® £4 ® {)4, = 0,00, —e40e4, and (c,) and (O') be the metric
tensors and the dual bases in V and И*, i.c., (c,,0J) — <5/, /,j = l,2,3,4, where <5/ is the Kro-
necker delta. Then a tensor ГсЛ' can be written as
The transposition ( ) and trace ( )z arc the linear mappings defined by the relations
(a ® b) ~b ® a, (b ® a) = a 0 b, (co 0 v)z = (v 0 co)t — (у,co), where ае/Г”, beTor, vGAr(T),
(ueA^L*).
B. Exterior and interior products
In the exterior algebra A( L*), the right interior product J by an r-vcctor v is defined by
the relation (Refs. 59-62) (u,v J co) = <v A u,w) for every ueA(L). It follows from the defi-
nition that v2 J (Vj J co) = (Vj A v2) J co. Let veAr( И), соеA5( K*). Then v J coeAv r( И*) if
r<s, v J co—(y,co) if and v J co = 0 if r>s. The interior product obeys the “anti-Leibnitz”
rule
v J (ш1 A w2) = (v J w1) Ao)2+ (-I)VA(vJ co2),
(2.2)
where veL, (Лл'(И, w2eA5(k*).
In the exterior algebra A(L), the right interior product L by a 5-form co is defined
similarly: (v L co,cr) = (у,co A cr) for every ctgA(L*), and (v L co1) L co2 — v L (w1 /\co2). Let
veAr( И), wgA5( F*). Then v LfoeAr_‘r( И) ifs<r, v L*>= (v,<u) if s=r, and v Lcj = 0 if s> r.
For arbitrary сое I7*, VjgA^ F), v2gA5( V), we also have
(vj A v2) L co — (vj L co) A v2+ ( — 1 )ZV; A (v2 Lco).
There exists a natural generalization of the exterior and interior products on some tensors
of type (r,s). The interior product 1 , the exterior product A, the double exterior product
A A, and the double interior product 1 1 of p-forms, ^-vectors, and tensors from Asr and Apq
are the bilinear mappings defined as follows:
ul (ul co)&v, (ca®v)l a=cj®(vl a), (2.4a)
(w0v)l (a®u)=co® (vl a)®u, (2.4b)
аЛ (cz> ® v) = (aAca) ® v, (co ® v) Au — ca® (vAu), (2.5)
(co ® v) A A (a® u) — (caAa) ® (vAu), (2.6)
(co®v)l 1 (u 0 a) — (u 0 cr)l 1 (co 0v) = (ul co) 0 (al v), (2.7)
where vGAr(F), ugA9(Iz), coeAs(K*), aeA^J7*), and
u J co, if c/Cs
ul co = col u = (2.8)
uL co, it q^s.
in case th' values of г, с/, and p are given, the symbol 1 in Eq. (2.4) can be replaced by J
or L in accordance with Eq. (2.8).
C. Operators (linear mappings)
Let vgA'V), саеАг(Е*), ГеЛ/. Then
ii — vl
<j~IA ш=ТыеЛ*(У*).
(2.9)
Thus, each tensor 7'<eA' naturally defines the linear mappings 7"':AV( F)-> A'( Ю and
7 AAr( I7*)-* A ( F*). fhe interior product Ту — ! |1 f'’2= 7\ 71 G/l/7 of the tensors, i t.ezl
and T2^A^ describes the composite mapping. We use here the operator notation for the linear
mappings and their compositions. We shall omit the symbol 1 if it does not cause a misun-
derstanding In particular, the interior product 7T • • -1 T of// copies of TciAs is denoted 7’".
By making use of the expansion (2.1), the relations (2.9) can be written in the coordinate
form
ls
(2.10a)
I,
(2.10b)
The tensors
O'1 A * * * A"0'v ® 0; A * * • A C;
'1
define the unit operators in the spaces of ^-vectors and s-forms [vU.=v, U/t> = co for every
vgAa(F) and сое Ал( F*)]. From Eqs. (2.6) and (2.11) we also have
=-L’,A AUh
1 •
1 1
LT3 —— UjA AU] A AU1( U4--U]A AU] A AU] A AU].
(2.12)
v
/
/
1
1
D. Tensors Ea and a”
Let
aeA^(F), veA5(F),
a^A^F*), weArF$).
The tensor
E* = aJ (aAUX-l)?I(UsAa) LctgJ/,
(2.14)
where r= 1,2,3,1, and the tensor
T>crA(aJ Ur) = (-l)f/($-/;)(U.v.Lcr) АаеЯД
where r= 1,2,3,4, r —#>0, s—p + r—q<A, have the properties
E”w = a J (ci Aro), vEj=(aAv) La,
]l“/y = a A (aJw), vI“ = aA (v Lcr).
Ifp = <7=l (s = r), then
E" + C=<a,cr>Ur,
- (Ea)2=<a,a)Ea, (3a)2= (a,cr)Ja.
(2.15)
(2.16a)
(2.16b)
(2.17a)
(2.17b)
Thus, if ае И, ere У*, and (a,cr) = 1, then E" and define the projection operators in Лл( И)
and Ar( F*). In this case
E. Dot product: Raising and lowering of indices
Let the last index of a tensor A be contravariant (covariant), let the first index of a tensor
В be covariant (contravariant). The dot product A * В is the contraction of A ® В on the
above-mentioned pair of indices. For example,
4
V • Ci)~CO • v = (v,<j) = X
/=1
(A®v) • (tu®z2) = <v,w)/i®z2,
(2.19)
where ve К, сое F*, and , Z2 are tensors of arbitrary types. Using the same notation “ • ” for
the scalar product, we can extend the dot product for tensors of any types. In particular, for v,
ае V we have v • a== v • g • a = 24A.= j ulgikak.
Consider v=X4_1 p'e, and = The relations
g-ez={V, /=1,2,3, g-e4=—H4, (2.20a)
4
u=g • v = u1t)1 + Л}2-Ьр3О3 — Л)4 = У vfi‘, (2.20b)
t i
g^-O^e,, z= 1,2,3, g-1-04=-c4, . (2.21a)
4
co=g~1 • rj = 6.)1e1 -{-6?2e2+ 6)363 — 6)464 = X
/=1
(2.21b)
describe the operations of lowering and raising of indices. In this article, the ^-vectors and the
s-forms (5-= 1,2,3,4) related by these operations are denoted by the same letters. The s-vector
will be distinguished from the corresponding 5-form by the bold type and the position of a label.
F. Star operators
The linear mappings *:AS( V) — A4-J( K) and *:AS(H*) — Л4-4(Г*) are defined by the
relations59,61
*y— —fl Lu, *6) = wJfl, , (2.22)
where vgAs(K), соеЛ^И*), П^'Л^Л^Л-»4, П = -e,Ле2Ле3Ле4.
For ??-vectors v, u and s-forms co, cr we have
(*v,*6j) — (v,w), (2.23a)
v A *u= (v,m) (—fl), w A *cr= (cr/o)fl, (2.23b)
**v—— (— 1 )5v, **6)= — (— 1 )Vu, 5= 1,2,3. (2.23c)
The star operator is described by the tensors
*U! =d2 ЛО3 Л О4 ® б! — ч)1 Л1)3 Л-04 0 еэ + О1 A h2 AO4 0 ез + О1 Аг)2 A-O3 0 e4,
(2.24a)
*U2=x)3Ai)4 0 Cj A c2 —i)2 A « C] A e3 —A -O3 0 С] Л + А г)4 0 e2 Л ез + 'Э1 A il3 0 e2 A e4
— лЗ1 A1)2 0 e3 A e4, (2.24b)
*U3 = 04 0 cq Ae2 A e3 + x)3 0 C! A e2 Ae4 —-b2 ® e! Ae3 A e4-F'&1 0 e2 Ле3 Ae4. (2.24c)
Let wgAv(F*), vgA4~s(F), 5= 1,2,3. Then
*U/u = *co, v*Uy = — ( — 1) A*v.
(2.25)
From the definitions of mappings A, J ,
(2.23)-(2.25) follow the relations
L, 1 , *, and formulas (2.14), (2.15), and
*(crAw) — ( — 1 )'aJ *6), *(aAv) — ( — 1 )v(*v) La, (2.26a)
*(a J co) — — ( — 1 )v(7 A *co, *(v L cr) = — (— 1 )\r A *v, (2.26b)
ид *Ц{=*1Ц Uy=*Uv, *ид *Ur=-(-l)TJr, (2.26c)
*US1 E” = jl'fl *ЦУ, E^L *иг-*ПД H'f, (2,26d)
where яе И, ere vgAa( К), шеЛ5( V^), E“ g Л/, e Arr, r+s=A^ 1,2,3.
G. Pseudoinverse tensors
In accordance with Eq. (2.9), a tensor T eA} ~ У* ® У defines operators in V and И*. If
det T~ (1/4!) (ГЛЛГЛЛГЛ A T)t /-0, the inverse operators in И and И* are defined by
the tensor
_ j Ill (ТА АТЛАГ)! П
= ~4 ('ГЛ ЛТЛ ЛГЛ ЛГ), ’
(2.27)
which satisfies the relations T 1 T ~TT
Consider now a tensor Ae К* ® V such that dim Ker A — s <4, i.c., there exist azG И and
V* (/ — 1,2,...,5) for which a,A =0, A{3'=0, a} Л • • • Ла5т^0, /31 A • * • A/JV=0. Let us supple-
ment aH...,as and /31,...# to two pairs of dual bases (a,), (af) and (b,), (/?), i.e., a, J a7
=bzJ Д7 = <5/, i,j~ 1,2,3,4. Then A can be written as
4
А— У zf/a;®b7,
/,y=s+ I
(2.28)
where the (4—s) X (4 —s)-matrix (zf/) has the inverse matrix (Л/). Therefore, the tensor
i
A ~ = E ® aj
i,j~x+ 1
has the properties
4 v
ЛЯ~=:Рде= X У a'®a/f
1=5+1 /= 1
4 v
A~A = V/j= X P'eb^Ui- X Р'ыЬ,,
i—s+ 1 /= 1
(2.29)
(2.30a)
(2.30b)
where Pa and P^ are projection operators (Рп2--Рд, Рд2 = Р^), and (Pa)f—(F/j)r —4—s, P(/l
=АРр=А, P^A~ ~A~Pa=A~. The tensor A~ is called a pseudoinverse (to A) tensor. It is
obvious that there exists an infinite set of pseudoinverse tensors with different Pa and P^. In
this article, we use several types of pseudoinverse operators [sec Secs. IV В, VII A, IX A, and
the Appendix],
III. BASIC EQUATIONS
The vector xeVf which locates an event with position r and time Z in an inertial reference
frame with an orthonormal basis (ej, can be written as x = 24s=1 х'е^г+лЛ^, where
r = 2/=1 x'ez, x4 = ct, and c is the velocity of light in vacuo. Therefore, the equation r • q=vot
q-f0, defining a uniformly moving plane interface between two media, takes the covariant form
x • Q=x J Q — fo, where
Q = q+ae4, Q=g • Q = # —czd4, a — u^c,
(3.1)
q and v0 are the unit normal and the velocity of the boundary, and is a constant scalar
parameter. In this article, we consider a multilayer structure which consists of linear media
uniformly moving with different velocities with respect to the laboratory frame. Let all n
boundaries have equal velocities and are defined by the equations xAJQ = fk, A—1,2,...,л,
where fi <£2 <•”<£«•
By applying Stokes’ theorem59,61 to the relativistic Maxwell equations dF~ 0, d*G — 0,
one can find the boundary conditions
QA (F(1)—T?(2))=0,
(3.2a)
Q J (G(l>-G’(2))=0.
(3.2b)
In the presence of surface charges and currents, the condition (3.2b) is replaced by Q J (<7 '
— Gf2)) =4rr/,/c, where QJQ=1, and J" is the four-dimensional surface current density
(QJF=0).
At oblique incidence onto the multilayer structure, a plane harmonic wave excites an
electromagnetic field of the form
F=F(£)exp(/x J r),
(3.3a)
G=G(£)exp(/x J t),
(3.3b)
where f = x J Q, and tgP is some given parameter such that QAtt^O. For this field, the
Maxwell equations reduce to
d
К
Q A F 4- ir A F = 0,
(3.4a)
0
— Q J (7+ir J G— 0.
(3.4b)
In a homogeneous linear nondispersive medium, the field and induction two-forms Fand
G are related by the operator constitutive relation
G—MF,
where M is an antisymmetric tensor of type (2,2), which can be written as
(3.6)
l</< j <4, kJ- < /<4
IV. EVOLUTION OPERATORS
A. Amplitude subspace
Using the identities (*Q) L(QAF)~0, QJ (QJ(7)=0, and Eq. (2.26a), from Eqs.
(3.4) and (3.5) we obtain
SjJF —0, s2JF = 0,
(4.1)
where Sj —(QAr)Af, s2 —*(QAr). If two-vectors Sj and s2 are linearly independent, i.e.,
S~S] 0 s2 — s2 & sp/^0, Eqs. (4.1) define a four-dimensional subspace (amplitude subspace)
7^/?gA2( J7*) of the six-dimensional space of two-forms. Let ? and Г2 be arbitrary two-forms,
satisfying the condition (1/2)(T1 5)(S] J2) (s2 J1) —(S] J1) (s2,r2)-/=O, where T —
— Г0/1. Then the tensor
J/ = U2-2T1 S/(T1 S),
(4.2)
has the properties ..7 2 = /Z , J7 r — 4, s/ — 0, /- tl~0, i~ 1.2. Thus, ,7~ is a projection operator
onto the amplitude subspace 7 \ (ATF — F for any T’e7 r), and /! and t1 specify projection
directions.
B. Evolution equation
Let u be an arbitrary vector satisfying uJQ = l. Setting 1 and r—s~2 in Eqs.
(2.13)-(2.15), we obtain the projection operators
(^•3)
with the following properties
(Z,)2=Z,e/l22, /=1,2,3,4; Z1+Z2=Z3+Z4=L2,
(4.4a)
/1 /2 — /2/! — Z3/4 — /4/3 — 0.
(4.4b)
Adding equations obtained by interior multiplication of Eq. (3.4a) by u and Eq. (3.4b) by
(ЦМ12)~ Lu, and then using Eqs. (3.5), (4.1), (4.3), and (4.4), one can find the evolution
equation
(4.5)
where
JT = [ -E“+ (Z4MZ2) ~ (MKur-I>f) ].r,
(4.6)
Ж JT = jr6/t22, and (Z4MZ2)“ is a pseudoinverse operator (see Appendix). The
latter exists if A = Q J M L Q is of rank 3, i.e., А Л A A A AA=^0. This condition can also be
written in an equivalent scalar form (*u)J (АЛ ЛАЛ ЛА) L (*п)^Д0. One can show that, if
the condition is met, S=^0.
C. Evolution operators in a homogeneous medium
In a homogeneous medium, the general solution of Eq. (4.5) has the form
F(?)=cxp(/^JT)F(0),
(4.7)
where F(Q') is an arbitrary two-form from 7^F, i.e., 27~F(0) = F(0). Let us denote by xt and
x2 the four-dimensional radius vectors of two arbitrary points in the Minkowski space. By the
definition, the evolution operator J^(x) relates field values at these points as follows F(x2)
= J7r(x)F(x1), where x = x2 — xj. Taking into account the expressions (3.3), (4.1), and (4.7),
we obtain
5^(x) = exp(zx J т)АГ ехр(/^.Ж’).
(4.8)
Properties of AAA and depend on the values of Q, r, and M. Let £z (/ = l,...,iV<4) be
eigenvalues of the restriction 3F\ of AY to 7/'r. Let sz and nt be geometric and algebraic
multiplicities of £z. Using the Cayley-Hamilton theorem, we obtain the spectral expansions
N
/=1
(4.9a)
George N. Borzdov: Tensor technique in Minkowski space
3171
T " lcxp(/x J A7),
(4.9b)
where A7~r+£;Q, P, is a projection operator onto the invariant subspace 7 'i corresponding to
£z, Tj — 33 — is a nilpotent operator, and tj is its index of nilpotence. P, and Р,- have the
properties
(Pz),=dim7<=^, PiT^Tf^Ti,
(4.10a)
(4.10b)
P, = 0 for r,= l; P/'-’^O, P/'=0 forZz>l.
(4.10c)
It is convenient to classify the operators 33 by values of the invariants N, nh sif and Due
to the isomorphism of linear spaces having the same dimensions, we can apply here, after an
obvious modification of the notation, the classification of linear operators and the explicit
expressions for the projection operators P, obtained in Ref. 56. From the mathematical point of
view, there are fourteen types of operator 33 (see Table I in Ref. 56). However, there exist
some physical restrictions on 37' (Sec. V).
In the general case, the electromagnetic field
F(x)=5r(x)F(0),
(4.11)
where 37 (x) is defined by Eq. (4.9b), consists of JV partial waves
/ E"1 (i7)n \
F>(x)=cxp(/xJA'>) + X F'(0),
\ n I ‘ /
(4.12)
where F7(0) =PyF(0), and, for an arbitrary x, F7(x)GPj.
If here 7y=l, i.e., Py = 0, or tj> 1 and 33 F7(0) =£yF7(0), i.e., PyF7(0)=0, F7 of Eq.
(4.12) reduces to the eigen wave
F7(x)=exp(/xJ A7)F7(0).
(4.13)
In other cases the relation (4.12) describes a wave with linear, quadratic, or cubic amplitude
dependence on coordinates. If 33 does not have multiple eigenvalues (7V=4, Tj — 0,
j = 1,2,3,4), the general solution F consists of four partial eigenwaves F7 with different four-
dimensional wave vectors • A7 = r+£;-Q.
D. Boundary operators
Consider the interface between Ath and к4- 1th media with material tensors Mk and
Since /jF —u J (OAF) and (Q J G) are continuous at the interface [see Eq. (3.2)],
using Eqs. (3.5) and (4.4), we can relate the boundary values FvA) and F k of Fas follows:
(4.14a)
(4.14b)
where
% 1.Л = ^2 + 1Л ) ~ Wk - Mk+ , ),
(4.15)
and can be found from Eq. (4.15) by the replacement k<^>k-j-1. From Eqs. (4.1),
(4.14), and (4.15), it follows that the boundary operators have the properties F/ ktk + }^k+\,k
= ^л.+ 1Л3',м.+ 1 = и2> s(1A)^w.+ 1=s(1A'+1), S2^W+|=S2, where s!A) = (QAt)Ma, s2
= *(QAt).
The operators > (4.8) and ,k (4.15) make it possible to find an evolution operator
relating field values at two arbitrarily given points of the layered medium.
V. EIGENWAVES
A. Dispersion equation
For the eigenwave
F(x) —cxp(zx J A')jF(O), (5.1)
Eqs. (3.4) reduce to
kFF^O, (5.2a)
kJG-O, (5.2b)
where k~ r-J-£Q- It follows from Eq. (5.2a) that F can be written as
/’-ЛЛ/, (5.3)
where f—у J F is a polarization parameter of the wave, and v is an arbitrary vector satisfying
vJ&=l. Substitution of Eqs. (3.5) and (5.3) into Eq. (5.2b) gives
X/=0, (5.4a)
vJ/=0, (5.4b)
X = kJML/c. (5.4c)
The system of equations (3.5)'and (5.2) has nonzero solutions (F = k/\ f=/=0) if the kernal of
X is, at least, two-dimensional. Therefore, the wave one-form к satisfies the tensor dispersion
equation
yA AxA Ay = 0,
which can be written also in the equivalent scalar form
(*v) J (x A Ax A Ax) L (*u) =0.
(5.5b)
B. The spectrum (£y, of the operator 3F
An eigenvalue of FF (4.6) determines the one-form kJ — of the partial eigenwave
FJ (4.13). Substitution of k = r-\-^Q into Eqs. (5.4c) and (5.5b) gives x = b2^4-£/?+C, and
Д(т4-^О)=а4^4 + азё3+а2Ь2 + о1^ + ао=0.
(5.6)
where
/1=Q JA/LQ,
В{ = тЛ M LQ,
jB2 = Q JMLt,
C—t J M Lt,
(5.7)
a0=<7o(r) = (*v) J (СЛ ЛСЛ AC) L (*v), (5.8a)
a1=r/i(Q,r)=3(*v) J (5АЛСА AC) L (*i>), (5.8b)
a2 = a2(Q,r) = 3(*v) J (А Л AC Л ЛС+ВЛ ЛВВ. AC) L (*u), (5.8c)
a3 = o3(Q,-r) = (*v) J (6ЛЛ AAA Л C+ В A Л Bh A B) L (5.8d)
o4 = <74(Q,t) = 3(*v) J (ЛА ЛИЛ AC+JA ЛВЛ A A) L (*u), (5.8e)
and v J Q=0, v J r=l. Therefore, the eigenvalues (J = 1,...,7V) are the roots of the quartic
equation (5.6).
Using the dyad expansion of y> one can easily verify that у A Ay A Ay = 3!A(£)*£® *k,
and (уЛ ЛуЛ Л у),= (у,)3 —3y,(y2),+2(y3),= 3!A(&)k J к. Непсе, the trace of Eq. (5.5a)
yields the dispersion equation
Съ)3~Зу,(у2),+2(у3),=о
(5.9)
obtained, in the component form, by Polevoi and Rytov. However, this equation, unlike Eqs.
(5.5) and (5.6), has additional roots defined by kJA = 0, which do not have any physical
meaning.
C. Degeneracy conditions
If £ is a и-fold root of Eq. (5.6), A=r+£Q and Q satisfy the following equations
ao(k) = (*v) J (^A Ax A A y) L (*u) =0,
<7,(0,A) =0, /=1,...,и —1, я„(О,А)^0.
(5.10a)
(5.10b)
Substituting В (5.7) into (5.8b) and using Eqs. (2.4)-(2.8) and (5.5), we find that the
condition (Q,A) = 0, i.e., n>2, can also be written in two other equivalent forms: (kJ M L Q
+ Q J M L k) A Ax A Ax = 0, and W J Q=0, where
W = ?/l 1 (kJM)+g-1 ’ [(ЛШ)! 1 77],
77 = (*v) L (x А Ay) J (*f),
(5.11a)
(5.11b)
and the direction of W does not depend on v which is restricted only by v J k=^0, Thus, to find
values of Q and т ensuring the degeneracy of Ж (4.6) and (4.8), which exist for any given
solution kJ of Eq. (5.10a), one can use a procedure consisting of (i) determination of Q from
VV J Q = 0 at k=kj\ (ii) calculation of the multiple eigenvalue ^ = uJ k} and the parameter
r=/<J—gyQ providing a degeneracy of and ; (iii) calculation of <72(Q,A7) and a3(Q,A7)
to find fhe algebraic multiplicity zzy- of from Eq. (5.10b).
D. Г ^arizations of eigenwaves
We can classify у by the dimension of its kernal as follows:
X'A Лх#Д yA AxA Ax=0 (dim kerх~2); (5.12a)
Xt^O, уЛАу=0 (dimkerx = 3); (5.12b)
X — 0 (dim kcr x — 4).
(5.12c)
In the first case [see Eq. (5.12a)]f an amplitude subspace of the wave F (5.1) is one
dimensional. Exterior multiplication of Eq. (5.4a) by u yields v/7 —0,-where v — — u А у Av. In
accordance with Eq. (5.5), we have (see Appendix) v= —(1/2)Ш. (уЛЛу)1 Я =
where H is some two-vector. Multiplying v by c J П, where c is an arbitrary two-vector such
that г(сЛ1)^0, and omitting an inessential scalar coefficient, wc obtain the two-form
F = c(xA J fl which uniquely determines the polarization of the eigen wave.
In the second case [see Eq. (5.12b)], x is a dyad, and the polarization one-form /belongs
to the two-dimensional subspace defined by w^L/—0, where ну—nAv, п = С]у, and is an
arbitrary vector (c^t^O). In other words, when у A Ax —0» the wave propagates along an
isotropic (optical) axis and may have any polarization.
It follows from Eq. (5.2a) that there are no eigenwaves with four-dimensional amplitude
subspaces. Moreover, in most linear media, eigenwaves with three-dimensional amplitude sub-
spaces do not exist either,56 i.e., X'/O-
VI. WAVES WITH TWO-DIMENSIONAL AMPLITUDE SUBSPACE
A. Wave equation for the one-form 7—v J F
Let v be an arbitrary vector such that v J Q — 0, v J r— 1, let /—v J F. Using Eqs. (2.17),
(3.4), and (3.5), wc express F through / and then obtain the wave equation for /in a
homogeneous medium
d
F = tA/—/Q A^/,
a2 d
f—iB — f + C/—0,
dj
(6.1a)
(6.1b)
where Л, /Д and C are defined by Eq. (5.7), and v J/ —0. Thus, each solution/of the wave
equation (6.1b) uniquely determines a solution F, G—MF of the basic equations (3.4).
At incidence onto a semi-infinite anisotropic medium, the wave F (5.1) usually excites two
refracted partial eigenwaves with different wave vectors and different polarizations. In some
cases, when these wave vectors coincide, Voigt waves48,63 and inhomogeneous waves with a
linear amplitude dependence on coordinates48,56,64 may arise. The total refracted wave has a
two-dimensional amplitude subspace Vj- and can be written as
/(x) —exp(zx J т)ехр(/£А)/(0),
(6.2)
where an operator A and a projection operator I f onto V f have the properties IfN—NIf=N,
I(Ij)2, Iff—/ for any feFf. Substitution of/ (6.2) into (6.1) yields
F=rA/+QAA/,
AN2FBN+CIf^0.
(6.3a)
(6.3b)
It follows from Eqs. (5.4) and (6.3b) that eigenvalues of A are defined by the roots (j — 1,2)
of Eq. (5.6).
Since dim Ker N—2, there exist only three types of A and, accordingly, three types of wave
A —£]Pi + biPi> Si-Ль2»
(6.4a)
F = k] A f] (0)exp(7x J A1) Ч-A2 A /2(O)exp(/x J A2);
(6.4b)
(6.5a)
F — A1 A/(O)exp(Zx J A1);
(6.5b)
d,
(6.6a)
F= [k] (Q + /fAJ) Ac® d]/(0)exp(fx J A1).
(6.6b)
Here, p1=Z/-p2=(.V-^Z/)/(^1-f2), /'(0) =p7/(0), A>-r+^Q, dJe-0, dZz=d, I
—e, and /(0)=vJZ(0) is an arbitrary one-form from Vj- (Zy/(O) =/(0)). Thus, the non-
degenerate operator N (6.4a) describes the superposition of two partial waves with different
four-dimensional wave vectors ky—T+£yQ and different polarizations defined by the projec-
tion operators pj [piP2 = p2pi Pj2 = Pj> (p/)r=l> J = 1,2]. The degenerate operators N
(6.5a) and N (6.6a) describe the eigenwave F (6.5b) propagating along an isotropic axis and
the wave F (6.6b) with linear amplitude dependence on coordinates (f —x J Q), respectively.
B. Explicit expressions for N and lf
1. Nondegenerate operator N
Using the Caley-Hamilton theorem (N2 — (^ifrom Eq. (6.3b) we obtain
aN=bIf,
(6.7a)
(6.7b)
(6.7c)
where
b^&A-C,
(6.8a)
XP=^p-b = tfA J M LA', 7= 1,2. (6.8b)
Both one-forms A1, A2 and amplitudes /?1 = A1 A (0), F2 = k2 A/2(0) in Eq. (6.4b) are
linearly independent. [It follows from Eq. (5.2a) that linearly dependent amplitudes can be
written as Fx = CjA1 A A2 and F2 — C2AJ A A2, where C{ and C2 are some scalars.] It follows from
Eqs. (5.2)-(5.4) that a sufficient condition for the linear independence of F1 and F2 has the
form and (or) у2А!т^0. We now assume that this condition is met, and both partial
waves satisfy the condition (5.12a) as well. Polarizations of the waves are defined by (see Sec.
V D)
Z7'= [c, J (Y/ A A Y/) ] J П,
f‘ = v J F‘~ [c, J (Y/-A A Yz)] J *vt 7—1,2,
(6.9a)
(6.9b)
where c, are arbitrary two-vectors. From Eqs. (5.4a) and (6.7c), one finds that N—^f
~/ Ag|, where g| is some vector. Let c be an arbitrary vector such that cr//1—0, n = cY2y-0.
Then, from Eq. (6.7b), we have nZy—0, i.e., n is a normal to Vf, Hence, the amplitude
subspace Vf is now uniquely defined by the two-vector wy=nAv, and a projection operator
onto Vcan be written as
Zy= V j 4-//• wy = U] -4- (v J //) o n — (n J //) # v, (6.10)
where h! is an arbitrary two-form (wyJ h=\) specifying the direction of projection. Now,
when If is found, from Eq. (6.7) we obtain
N^cr}blfl (6.11a)
Л^ = (6.11b)
N=£2If-f{®c'x2If/eaf', (6.11c)
where xeTj and cf are arbitrary tensor and vector parameters. If det a^O, all these expres-
sions are equivalent. However, if det£/=.O, one must use the expression (6.11b) with x such
that det + or the expression (6.11c). It should be emphasized that the obtained
here operators If and N do not depend on values of auxiliary parameters cz, c, c', and x.
2. Degenerate operators N
Let now = i-e-> £1 is a double root of Eq. (5.6). In this case, If is defined by Eq.
(6.10), Wy=nAv, where n = c^z], ccq /3 = 0, ах — 2^\А + В, and the expression (6.11c) takes
the form
N— $\I f~\r С г) d,
e ® d— — fx 0 c'^q/ r/c'cf!/1.
(6.12a)
(6.12b)
If the wave propagates along an isotropic axis (^1 A A/j =0), [see Sec. V D], and N
(6.12a) reduces to N (6.5a). Otherwise, since Iffx~f\ Azi/3—0, the dyad e®d (6.12b) is a
nilpotent operator, i.e., N (6.12a) describes the wave F (6.6b).
C. Surface impedance operator
As before, let u and v be auxiliary vector parameters related with the one-forms Q and т
by the conditions
u J Q = v J r= 1,
11 J r=y J Q — 0.
From the boundary conditions (3.2), it follows that the one-forms:
bQJG,
are continuous at the moving boundary. Here, w=uAv and
/~v J F=y-FU)Q,
(6.13)
(6.14a)
(6.14b)
(6.15a)
(6.15b)
F w w -1 F.
We define a surface impedance у as a linear operator relating boundary values of the
one-forms h and <p so that
A = /ф,
Qy^r/^0,
(6.16a)
(6.16b)
yQ — yr=O,
(6.16c)
To find an explicit expression for y, let us set //=QAr/nJ Q in If (6.10) and introduce
the projection operators
/ —Uj + (Q Ar) • w —Uj — Q® u —г® v, (6.17a)
I'—gig 1 = + • (QA r) =Uj — и ® Q — u® r. (6.17b)
From Eqs. (4.1), (6.10), and (6.13)—(6.17), we have
Icp=<p, rh~ht I’y=yl = y, (6.18a)
If—cp, If<p=f, IfI=If. (6.18b)
Therefore, for the wave f (6.2), we can restore the two-form F (6.3a) from the one-form f or
cp as follows:
F^7rf=z7r(p,
(6.19a)
/Г-тА/у’+ОЛ^еЛг1. (6.19b)
Finally, using the relations (3.5), (6.14), (6.19), and the definition of the surface impedance
operator y, we obtain
y=QJM7/\
(6.20)
VII. BOUNDARY VALUE PROBLEMS
A. Interface between two moving media
Let us find reflection and transmission operators of the interface between two anisotropic
and gyrotropic homogeneous linear media with constitutive relations of the form (3.5). We
assume that, in the general case, both the media and the interface are uniformly moving with
different velocities with respect to the laboratory frame.
Let us denote by y, Nri yrt and Nd, yd the operator parameters of incident ф, reflected
cpr, and refracted (transmitted) yd total waves. They can be found from the expressions (6.11c)
[or, in case of degeneracy, from (6.5a) or (6.12)] and (6.20) for the corresponding media.
From the boundary conditions (3.2) and the definitions (6.14), (6.16), we have
<p + (pr=<pd, ycp + Y,ipr='Y(fp‘1.
(7.1)
For an operator у satisfying the conditions (6.16), there exists a pseudoinverse operator (see
Sec. II G)
y- =2(*wl yl *w)/[*w(yA Ay)*tu]
(7.2)
such that у у=Ц yy — I', ly ~y~ Г— y . Using the pseudoinversion, we immediately
obtain the solution
4>r=r(p, <pd=dq>, (7.3)
''=(Г<7-П) (Г-Zrf). d={'yd—'Yr} (y—Yr), (7.4)
where the reflection and transmission operators rand d have the properties /-\-r—d, Ir~rl -~-r,
Id~dl=d. The obtained relations are valid for the incident wave with any polarization state,
i.e., p is restricted only by the condition w = O (see (6.14b).
B. Multilayer structure
Consider now the multilayer structure described in Sec. III. Let /'"(xj, //r(X1), and Fd(x,)
be boundary values of two-forms F, Fr, and Fd of incident, reflected, and transmitted waves.
Using the evolution and boundary operators (see Sec. IV) and the surface impedance opera-
tors, we can relate the amplitudes
as follows;
where
<р=Е£Г(х1), <p''=E£F'(x„)
4>+4,r— aq>d, Y<p+Yr
a=E^7/Q,
0=QJM2,O7
__ ... cr <z?
’7 2, J 2,3 '7 zj'7 zi.h
exp[/(x,— x„) J r],
(7.5)
(7.6)
(7.7)
(7.8)
к=У k:exp[/(f*_!-^)jrA],
(7.9)
and 7/^d is the restoration operator [see Eq. (6.19)] of the transmitted wave F. Here, Y/ is a
characteristic operator of the multilayer structure, *s tbe coordinate of the A'th boundary,
and -^a» *^a> an^ ^A.A-t-i are defined by the expressions (4.2), (4.6), (4.8), (4.9), and
(4.15), where M—Mk is the material tensor of the Ath medium.
It is not difficult to show that Ia~al~a, ГР—/31=/3. Therefore, using a pseudoinverse
operator, we obtain the solution of Eqs. (7.6) in the form of the relations (7.3) with the
reflection and transmission operators
r—ad—I, d=(J3—Yra) (у-уг).
(7.10)
The operators r and d relate the one-forms pr, pd, and p (7.5). The reflection R and
transmission D operators, relating directly the two-forms Fr{xx\ Fd(xn), and F(x1), are
defined by the relations
Fr(X1)=/?F(X1),
F‘/(x„) = Z)F(x1),
(7.11)
Я=2/>Е£, Z>=7/SxZES,
(7.12)
where 7Kr and 7F~d are the restoration operators of the reflected and transmitted waves.
C. Energy-momentum tensor and radiation pressure
Using the interior and dot products, we can write the energy-momentum tensor1,42 of type
(1,1) as follows:
T=-(l/47r)[F-G+(l/2)(GJf’)U1].
(7.13)
Let К be the laboratory frame with dual bases (e,) and (O'). Since I3 = Ui — -fl4 ® e4 —г)1 ® e!
4--fl2 ® e24--fl3 ® e3 is a projection operator onto a three-dimensional Euclidean subspace,
— e4Tx>4, I3TI3, — ce4TI3, and НзГг>4/с are the energy density, the Maxwell stress tensor, the
Pointing vector, and the electromagnetic momentum density in K, respectively.
Consider now the uniformly moving interface between two media described in Sec. III. Its
four-dimensional normal Q (3.1) defines a three-dimensional hyperplane in the Minkowski
space. Moreover, using the definition of volume elements,1 one can see that cQ is exactly the
volume one-form of the three-dimensional element of this hyperplane, which is being swept out
by a unit interface element cr] in unit time as seen from K. On the other hand, the radiation
pressure force is equal to the three-dimensional momentum obtained by <7| in unit time in
the frame K. Hence, in accordance with the definition of the energy-momentum tensor, we have
I3(T + T- Q=I3 (P+ P-P1),
(7.14)
where T, Tr, and Td are the energy-momentum tensors of the incident, reflected, and trans-
mitted waves, and P= TQ, Pr=TrQ, and Pd= TjQ. In other words, the one-form <5^ of the
radiation pressure in К is the spatial part of the four-momentum P^-Pr— Pi. This approach can
also be applied to uniformly moving multilayer structures and reflectors (see also Sec. VI-
II A 7). In the latter case, the transmitted wave is absent. Therefore, the pressure on the
reflector is given by
>^Ц3(Т + Л)О-К3(Р-1-Рг).
(7.15)
It should be noted that the relations (7.13)-(7.15) are written for the instantaneous real
values. The expressions for the average values follow from them straightforwardly (for in-
stance, see Sec. VIII A 7).
VIII. SOME APPLICATIONS
A. Reflection and refraction of waves at the interface between two moving isotropic
media
The reflection and transmission of electromagnetic waves by a moving semi-infinite dielec-
tric medium18’32 and a moving interface between two stationary media11,21,26,27 have been
intensively studied in the literature. The following cases of the movement have been investi-
gated: (1) the dielectric medium moving perpendicular to the interface;18,19,21,22,26,27,31,32 (2)
the dielectric medium moving parallel to the interface so that the plane of incidence is paral-
lel,18,19,25,27,28’29 perpendicular,23 or arbitrarily oriented20,24,30 with respect to the velocity. How-
ever, most of these investigations were confined to the special cases when the incident wave
propagates in the vacuum18 2O-22-23'28’3O’3I>32 or jn a stationary dielectric medium,11,21,25,29 and its
electric field vector is parallel11,19’21,24,28,32 or perpendicular11,18,19,22’25,32 to the plane of inci-
dence. The interaction of E waves and H waves with the interface between two isotropic media
moving with different velocities which are either perpendicular or parallel to the interface has
been investigated in some detail by Bolotovskii and Stolyrov.26,27 In this section, we study the
reflection-transmission problem in the general case when the velocities of the dielectric media
and the interface between them, the orientation of the plane of incidence, and the polarization
of the incident wave are arbitrary.
V. EircnvA'jvrr in a moving irotropic medium
Che field and induction tensors can be written as F — В J *г>44-2ГЛ 01 and G~H J *04
H-jDAxV, where
£=-p4JF, Z>=--e4J(7, B = *e4LF, H = *e4LG
(8.1)
are the electric and magnetic fields in a Lorentz frame К with dual bases (c,) and (aT). Hence,
the material tensor M (3.5) of an isotropic medium which is at rest in a frame K' with bases
(e/) and ('9J ) have the form
M=- E*X + eC S 7- 5 Л A 5,
/.i o4 * * * a4 2[i
(8.2)
where
5 = и1+Л1')4'®е4. = Ъ|+л„(-/? + {)4) ® (Д + е4), (8.3)
e and 'i are the permittivity and permeability of the medium in K', (3=vm/c = c4>/(e4' J f)4)
— e4 is its normalized velocity with respect to the laboratory frame K, /3 —g • Д, A — 1, and
Л0=Л/(1-/32 *).
Since (SA KS){kKf)^2(SkKSf), from Eqs. (5.2)-(5.4), (7.13), and (8.2) we im-
mediately obtain the orthogonality condition for f, the dispersion equation, and the energy-
momentum tensor
k5/=0,
(8.4a)
k5fc= (<u/c)2[m2 — 1 -A„( 1 — tn • Д)2] =0.
(8.4b)
7’=(l/47r^)(f5/)A:®kS’
= ( 1/4Я/1) [E2—Л„(Д • E)2] (/n —О4) ® [ш-|-е4-|-Л„( 1 — m • (3) (/? + c4) ], (8.5)
where к — / 4- (o>/c)c4 — (tu/c)(ni 4- e4). Here, the cigcnwave is characterized in AT by the
electric field strength E [F—k/\ f = (/л/ — О4) Л E], the three-dimensional wave vector Z = kl3,
the frequency tu — ck J d4, the refraction vector m = , the refractive index and the wave
normal n0 (n2 = 1), where xl4® е4='Э1 ® e! 4-'32 ® e2 + xP ® e3. Let 0 be the angle be-
tween no and Д, i.e., m • l3=n/3 cos ф with [3—vtn/c. Then, from Eqs. (8.4b) and (8.5) we find
the refractive index the phase and group velocities and vg in К
с — A/3 cos [ (1 —/32) (eji —Д2 —ЛД2 cos2 i/0 ]1/2
vp~" cos2
6/j “ 1
Uf” cos
/J
(p- — 1)2
c4TI3 m + 4( 1 — m Д)Д
V«=C е4Г-04=С 1+Л0(1-т-Д)
1/2
(8.6)
(8.7)
(8.8)
with m = «no. The relations (8.4b) and (8.6)-(8.8) are equivalent to the relations obtained by
Bolotovskii and Stolyrov,14,27 Kunz11 and Saca17 on the basis of different methods.
2. Laws of reflection and refraction
Consider two semi-infinite isotropic media with the interface normal Q (3.1). Let j3,=v,/c
be the normalized velocity of the /th medium with respect to K, let and /.q be the permittivity
and the permeability of the medium in its rest frame, i— 1,2. The four-dimensional wave vectors
of the incident k, reflected kr, and refracted kf/ waves differ from each other only by Q
components. Therefore, given r, we obtain the solution
= r + Q [ - rS,Q ± (r^ ,-r)1/2 ]/(QS,Q) (8.9)
of the dispersion equation =0, where
S, = U1+A10(-^ + '94) ® (/?, + e4),
(8.10a)
A,0=(^,-l)/(l-/32), (8.10b)
^,=S,Q®Q5,-(QS,Q)5/>
(8.11)
and the determinant tensor satisfies the identities QzZz=0, If the incident wave
propagates from the first (second) medium, then k = k1+ (k = k2_), kr=kj_ (kr=k2+), and
k^=k2+ )• consider here the first case, but it is still profitable to use the subscript
in some formulas.
Let us now formulate the obtained covariant laws of reflection and refraction (8.9) in
terms of the laboratory frame K. Let us direct ei and e3 along the plane of incidence and the
interface normal q, respectively, i.e., Q — e3-hcre4, a — vjc, and the wave vector of the incident
wave takes the form
Z = (zu/c)m = (zu/c)n(sin 6 G] 4-cos 6 e3).
(8.12)
Given the angle of incidence 0, one can find the refractive index n from Eq. (8.6) with
Л—Aj — Cj/i] — 1, /?i, ft cos ip—sin 04-/?i3 cos 0, By substituting т = к
— Z 4- (o»/c)e4 into Eq. (8.9) and separating the time and space components of k]_ and k2+
in Kt we find the frequencies cori cod, the wave vectors Zr, Xd as well as the refractive indices
nr, nd and the reflection 0r and refraction 0d angles
a)j = co( 1 4-«Vy), (8.13a)
Zy= (fu7/c)m7-“Z4- (a>/c)v7e3, (8.13b)
л —- (m2) I/2= (/r4-2?/v, cos 0 + v2)1/2/( 1 4-«v.-), ' (8.14)
J •'J
cos 0j~ (\/n ,)m7 • e3— (// cos 0 +v7)/(zz24-2nv7- cos 04-v2)1/2, (8.15)
where j—r, d\ vr~ — vr/=[ — x24- (x2 — cr2<5) l/2]/cr2 with
cr, = Q5ZQ = 1 — nr - A/w(fit3 - a)2, (8.16a)
;q= (m + e4)5£) = z7 cos 0 — a-pAJ0( 1 — /3,m) (J3l3 — a),
(8.16b)
(S—(m4-с4)52(ш — d4) ==>r — 1—Л20( 1-~/32 • m)2. . (8.16c)
If the first medium is (1) the vacuum zz=l, Л^Лю —0) or (2) a stationary
dielectric (/?, =0, — Л] — Л10=лг — 1), and the second medium moves (a) perpendic-
ular to the interface (Д21 =/322 = 0, a—/?23) or (b) parallel to the interface and the plane of
incidence (cz~/322 —/323 = O) or (c) parallel to the interface and at an arbitrary angle to the
plane of incidence (rz~--/L3~-0), the general expressions (8.13) and (8.15) reduce to the
relations obtained by Yeh’M9 [for the cases la and lb], Pyati20 (1c), Tsai and Auld21 (2a),
Daly and Gruenbcrg22 (la), Mukherjee25 (2b), Chuang and Ko’1’ (1c). For the moving
interface between two stationary dielectrics (/?]— /?2~0), the expressions (8.13)-(8.15) re-
duce to the relations obtained by Kunz,11 and Bolotovskii and Stolyrov.26,27
The solution (8.13)—(8.15) is written in terms of given co and 0. Consider now another
very useful formulation of the reflection and refraction laws. It is based on the use of
г=:74(лс14-e4), where r4 and 5 are some given scalar parameters. In this case, from Eq. (8.9)
we obtain the frequencies, the refraction, and wave vectors of the incident, reflected, and
refracted waves in the form
(8.17a)
coi± = cr4 (1 + avi±),
(8.17b)
where
v,± = [а+Л,о(^/3,1-1)(Д3-а) ±7?,]/<7,-( (8.18a)
7?2 = 172(^) = (т^,т)/(74)2 = 1/2(е//,+Л„/3;7)+Л,/-. (8.18b)
t/2=1 — s2+s2a\ (8.18c)
Wc shall apply these relations to find the conditions for some special cases of reflection. Such
conditions are usually formulated in terms of the angle of incidence 0 (maximum, crirical,
Brewster’s angles). However, it is more profitable to use for this purpose the refraction vector
m which uniquely determines the direction of propagation (including 0, of course,
cos 0 = m-e3//z) as well as refractive index n (л2 = т2) of the incident wave.
If /?j = 0, then /7 = (ej/ij)1/2, т1 + = и (sin 0 Cj +cos 0 e3), and the parameters 5, v1 + , and
0 are related as follows:
s~n sin 0/( 1 —an cos 0),
v1 + = n cos 0/( 1 — an cos 0).
(8.19a)
(8.19b)
5. Maximum and critical angles of incidence
The maximum angle of incidence (Ref. 11) 0t/n describes the case in which the wave
vectors of the reflected and incident waves coincide, i.e., kr=k. The corresponding refraction
vectors mi + —of these waves arc determined by the expressions (8.17b) at i— 1,
7]]=0, and 5=5i±, where
Л,Л1 (afa-1) ± 41 +A,0( 1 -/3?t -/З-з) ] [ 1 -а2-Л,„(/3,3-а)2]
1 -а2-Л,0[/32( 1 -а2) + (/3,з — а)2 ]
(8.20)
arc the roots of the equation t]2(5)=0. Thus, there exist two such vectors in any plane of
incidence. In particular, for the moving interface of two stationary dielectrics = 0), we
have m1/M= cz261jit1)]1/2e14-a6I^1e3 and cos 0h„ = a(61/z1)1/2. The latter coincides
with the expression obtained by Kunz.11
If
7)2 (-V) =7/o(e#^2 +^20^22^ T^20Z2^^’
(8.21)
the refracted wave becomes an evanescent wave with complex propagation constant v2+
(8.18a). Hence, 52 + and 52_ (8.20) arc two critical values of 5, and the corresponding critical
values of the refraction vectors of the incident m1 + , reflected mt_, and refracted m2+ waves arc
determined by Eq. (8.17b) at .v=.y2=t (8.20) and r/j = (j2 ±) (8.18b). For the cases treated
earlier in the literature, namely, (a) 0, /322 — p23~a — Mi=Zz2“l (Shiozawa and Kum-
agai,29 Srivastava29); (b) Д^О, /323 = a = O, Mi==Z12=1 (Shiozawa and Hazama24); (с) /3}
=p^ = 0 (Kunz11), these expressions for the critical values reduce to the relations obtained by
different methods.
4. Operator parameters of plane waves in isotropic media
Let us find the operators If, N, y, and 7Г (see Sec. VI) of a plane wave in an isotropic
medium. Substituting M (8.2) into Eq, (5.4c) and using the dispersion equation (8.4b) and
identities
{AN KB} L0-M0) ЛБ+(^)ЛЛ, eJ (§ KA} - (e J $}A® {eA), (8.22)
where A,BeT}\ eeV, Def* we have y= — Sk& kS/p. Hence, у satisfies the condition of
isotropic axis (5.12b), and n —k5 is an amplitude normal [see Eq. (8.4a)]. Substitution of M
(8.2). n = k5, and hJ — QAk/nJQ into Eqs. (6.5a), (6.10), (6.19b), and (6.20) yields
Z/=U1-k®v-Q^k5/kS’Q, N^If, (8.23)
7Г=кKIf=kAUi-kAQ^k5/k5Q, . (8.24)
1
у=д [(k5Q)5-5k®Q5](U1-Q®k5/kS'Q), (8.25)
where £ = u J k. The polarization one-forms f and cp of the wave are related as follows [see Eqs.
(6.15)’ (8.4a) and (8.25)]:
/=(Ui-Q$k5/k5Q)^,
fS/=M(^y<p)/(k5Q).
(8.26a)
(8.26b)
Since kZ45fQ~ ± (rZ \t)1/2 [see Eq. (8.9)], using the expansion k—т+^Q and the relations
(6.16b) and (6.16c), we can rewrite 7 (8.25) as
where
Here, 'уа, and a, b are the eigenvalues and the eigenvectors of у (a J b = 0, ay—yaa, уа = уоа,
by—ybb, уЬ = уьЬ), the projection operators Io, cra, ab satisfy the relations oacrb — crb(ja — 0,
Io = сгa-\-ch, & and A'0 are defined by Eq. (8.11) with St=S and respectively. The
impedances у and y,. of ihc incident and reflected waves differ from each other only by signs.
3184
George N. Borzdov: Tensor technique in Minkowski space
5. Reflection and transmission operators
Let us find now the reflection and transmission operators of the interface between two
moving isotropic media described in Sec. VIII A 2. Substituting Q —Сз + «е4, + c4),
Sz (8.10a), and Xz), (8.11) into Eqs. (8.27) and (8.28), we obtain the impedance of the 7th
medium
where — ® e2, e0 = et 4-5cte34-sc4 and
7,o=> °ia=« a,/a, J a‘,
Tib=, <?ib= b'® b,/b,J b‘>
*^i bi'lS-o »
b,=^A„-fee2.
(8.29)
(8.30a)
(8.30b)
Let us set w = Q A r/[(Q A r) J (Q A r)] as the auxiliary parameter in Eq. (6.17), i.e.,
1—Г=1о. Finally, since y~ —yr=yl and Yd— y2, from Eqs. (7.4) and (8.29) we easily obtain
an explicit formula for the transmission and reflection operators d and r
^г+/й=2(/о+Г)- = 2[(1 + Г,)/0-Г]/(1 + Гг+Г,),
(8.31)
where
7
a ® a2
(8.32)
is the interface impedance (relative impedance of the second medium), and
^10^20
ephPiVi
(/?i2z2—-/J22/’ j )2*
(8.33a)
Г, = ( 1/2) ( Г Л Л Г), - Г | f2 = YlaYl l/Y \«Y\b = P\ |/Z2
(8.33b)
are its invariants. Here, Г12 = Г/2± [(Г/2)2—-rj1/2 are the eigenvalues of Г. It is essential
that Г,= Г1Г2 is independent of the velocities of the media and the interface.
Since the dyads pk~ (Г — Г3_^/О)/(ГЛ — Г3_А) (£=1,2) are projection operators with
the properties p2 = pk, PkPi-k — ®’ (РкЬ=^> P\ + Pi~Io> the tensors Г, r, and d can be written
as
Г = Г1Р1 + Г2р2,
(8.34a)
(8.34b)
(8.34c)
where rk— (1 — ГА.)/( 1 + Г\) and —2/( 1 + ГА) arc the reflection and transmission coeffi-
cients for the incident wave with amplitude + =(pk which is an eigenvector of Г, i.e., Гср1 +
= Гk(p] ", <px~ = rk(px" , and <p2+ ~dk(p[ + . Such eigcnpolarizations arc defined by the one-forms
1,2,
(8.35)
where Co is an arbitrary scalar amplitude, and polarization parameters
—<APl+^2p2>
3185
2 "^2
— (бэ+ЛэоА/(8.36a)
P2~ — +^'20Г2^22/М27?2
(8.36b)
are found from Eq. (7.1) by substituting у,- (8.29) into Eq. (7.1b) and then multiplying the
latter by e2.
The obtained covariant solutions are formulated in terms of the one-form q> (6.14b). Since
F — k/\ /, G~Sk /\Sf /p, and f — —Q(k5<jp/k5Q), the field and induction tensors as well as
the field vectors E, В, H, D (8.1) in the laboratory frame К can be easily restored from cp. In
particular, consider the parametrization applied above (7—Г =1o). Let E"1 + , Ex~, and be
the electric field (in K) one-forms of the incident, reflected, and refracted waves, i.e.,
= (^±_^4) AFZ±. Then, from Eqs. (6.14b), (8.1), and (8.17), one finds that <?'* and E^
are uniquely related as follows:
<p/± = (c/«,;h) [(1/^)(£',±+5аГ3±)(-Э1+^3-^4)+4±^2]. (8.37)
7"’/±=134-/пг± ®e4-x/±(^3-a/n'±) ® b,,
where
X/± = ± A/o [ (1 - or) (sfi,! - 1) + (a ± ту,) (Дз - a) ] /стр/,-.
(8,38a)
(8.38b)
(8.39)
Since cpx~ — rq)X + ' cp2+—dqi+ with rt d (8.31) or r, d (8.34), the solution of the reflection-
transmission problem in terms of the electric field strength in К has the form
, iTntvi
E} =--------— 7. rF1’’, (8.40a)
l+avl +
l+av->_+_
£•2+ =------7Л dE^. (8.40b)
l-havH. +
From Eqs. (8.29) and (8.30), one finds that and y2 commutate only if
(б|/А| — 1 ) (cqii — 1 ) [/^12 — [32\ —-УОг/Зэз) —Pl2 (5 — 1 —5CC/?]3) ] —0.
(8.41)
If [3n=f322=0 or one of the media is free space (qz,= 1), this condition is satisfied identically.
Otherwise, given the plane of incidence, there exists only one value of 5 for which and y2
commutate. When the condition (8.41) is met, the general solutions (8.34b), (8.34c), and
(8.40) simplify drastically because then F| — y2o/y u== ^7/2/77^2, Е2 = у2Ь/у[Ь=7][€2/бх'Г]2,
Pi — aXa — a2tli p2~(J\b~(72b' In particular, it is satisfied identically in all boundary value prob-
lems treated in Refs. 11, 18-23, 25, 30, 32. To obtain from Eqs. (8.34) and (8.40) the
amplitude relations found in these articles, it is sufficient to impose the corresponding restric-
tions on the polarization of the incident wave, the velocities, and the material parameters. For
instance, at F1 + = Eo$\ £i=0, /?21 =/?22 = 0, /?23 = a, we have
(?/l+?/2)U -1-CZY/i )
(8.42)
where 7/2—l ™52-|-л2сг2, 7/2 = e2 —52 + л'2а2. Written in terms of the incidence angle 0 (s
— sin 0/(1 —a cos 0)), the relations (8.42) coincide with the relations obtained by Daly and
Gruenberg.22
6. Total transmission (Generalized Brewster's condition)
We observe from Eq. (8.34) that the reflected wave vanishes only if 1 = cpA (8.35) and
Г;. — 1 (t^1 = r<pl + = = dk — 1, 1, cp‘ 4 — qj11 ~-cpk). Since Г\ — 1 results in Г,— 1
+ f\, from Eq. (8.33) we immediately obtain a generalized Brewster’s condition
(fl772 “7?le2) ( 77 lM2 —1т?2 ) 10^20(^12^2 “/^22Z1 )2’
(8.43)
which is a quartic equation with respect to .v [sec Eq. (8.18)]. Let л;, be one of its roots. The
expressions (8.17) and (8.35) with s~so yield all parameters of the incident, reflected, and
transmitted waves for the case of total transmission, namely, the Doppler factors co}_/co[+f
co2 , the refraction vectors nt] t, m2 M and the amplitudes q^ 1 —<p2" — tpk (the reflected
wave has, of course, zero amplitude). The relations (8.38) with s~s„ and <fl 4 = <p2 1 =<pk
(8.35) yield also the electric and magnetic field vectors E/ + and — ш,XE/+ of the inci-
dent (z=l) and transmitted (/=2) waves in the laboratory frame K.
There exist two cases (i) /ЦсЛго” 0 (a vacuum-dielectric interface) and (ii) fin=fiu = Q
in which the right-hand side of Eq. (8.43) vanishes identically, and the quartic equation (8.43)
splits into two independent quadratic equations with respect to .v
(8.44a)
??le2>
(8.44b)
which correspond to the different cigenpolarizations (sec Sec. VIII A 5), namely, Eqs. (8.44a)
and (8.44b) result in Г] = 1, /3—0, and Г2= 1, r2 —0, respectively. The Brewster-angle phe-
nomenon for isotropic dielectrics has been intensively studied by various au-
thors,11’19'20’25’26’i(’’31 but their investigations were confined to the special cases (i) (Refs. 19, 20,
26, 30, 31) and (ii) (Refs. 11, 25) with some additional restrictions on the velocities and the
material parameters. Taking into account the corresponding restrictions, from Eqs. (8.17b),
(8.35), and (8.44) one can easily find the refraction vectors and the polarizations of the
Brewster’s waves for all these cases.
7. Radiation pressure force
For the interface between two moving isotropic media (see Sec. VIII A 2), the average
radiation pressure force is given by
^ = ^[(?t+W‘ + )/l + -(<ptW1 )^l--(^2 + ?2<₽2 )A+1.
(8.45)
where + , ^1-=r<p1+, cp2+ =dcpx + are the amplitudes of the incident, reflected, and trans-
mitted waves, the wave vectors Z, b, the impedances yz-, the reflection r and transmission d
operators are defined by Eqs. (8.17), (8.29), and (8.31), respectively, and 7?* is the complex
conjugate of cp. The expression (8.45) directly follows from Eqs. (7.14), (8.5), and (8.26b).
For the isotropic media (y=—]'г=уи Eqs. (7.1) yield the identity cp^^Ticp2^
— <p*+yl<p1 + — уiсрTherefore, substituting Z;± (8.17) into Eq. (8.45), we obtain
(8.46)
where
^/±='r4(¥’,*±W'±)/87r
= Wj±7/,/{ (1 + avu)2 [ 1 +л/и( 1 - m,± • /?,) ]}, (8.47)
Ro—A}_/A]+t and tu, is the energy density of the wave with the refraction vector m. .
(8.17b).
The obtained general formula is valid for some arbitrary angle and plane of incidence,
polarization of the incident wave, velocities of the media and the interface. At any values of
these parameters, the force .5* ? is directed along the interface normal. Let the direction of
propagation and the energy density of the incident wave be fixed, so that we study the polar-
ization dependence of Y/~p. Then Ro is the only parameter in Eq. (8.46) depending on the
polarization. In particular, for an eigenpolarization of the incident wave [<p14 =cpk (8.35)] we
have cp' — rk<px and R()=rf~ (1 — Гл.)2/( 1 + ГА)2, When, besides, the condition (8.41) is
met, ^2 = Y2i/Yib=:Vi^iV2f and
^o=(t7iM2-Mi'j72)2/(7/iM2+MP?2)2, (8.48a)
= (617/2-'??1^2)2/(6l7/2 + 7?le2)2’
if b A Z?! =0.
(8.48b)
In the special case when the wave is incident from a vacuum on a dielectric half space moving
normal to the interface (6j=/i2 = 1, Z?23 = a), the formula (8.46) with Ro
(8.48a) and R() (8.48b) reduces to the expressions obtained by Daly and Gruenberg,22 and
Kalluri and Shrivastava32 for E wave and H wave, respectively.
Similarly, from Eqs. (7.15), (8.5), and (8.26b), we obtain the average pressure force on a
moving reflector
+ )^i-l
=J)+ [.s(l-A„)C| + (v1 + -Aovl_)e3]
"’i । 7?l[.v(l-7?n)C| + (vi + -AcV|_)e3]
(1 -1-cti’n )5l 1 -l-Aiot 1 — ni l( •/?,)]
(8.49)
For an ideal reflector (Ro— 1) moving in a vacuum (Cj — 1), from Eqs. (8.10), (8.18), and
(8.19) we have Zl0=0, iq± = (a±7/j)/(1 — a2), iq —• (cos 0 — a)/( 1 — a cos 0), and the ex-
pression (8.49) reduces to the well-known Einstein’s formula65
2т/|(1—cr2) 2(cos£ — a)2
p~w\ + /1 : ту ез~ ; 2 ез •
1 r (l-haiqL 1—a
(8.50)
B. Degeneracy of the characteristic operator of biaxially anisotropic layer
The material tensor of a nonmagnetic anisotropic medium has the form
з
л/=и2--а4л X (с,-1){)'«с,лс4>
(8.51)
where (e,) and (O') are dual bases of the rest frame A', and 61>€2>e3 are the eigenvalues of
dielectric permittivity tensor e = eje| ® г)1 Тб2е2® г)2-*-63e30О . Consider the incidence of a
wave with the four-dimensional wave vector к onto an anisotropic layer cut so that Q = e2. Let
the parameters r, u, and t1 (see Secs. IV A and IV B) be given as follows: r —k —A2e2
= ^(m + c4), u — c2, ?=d2Ad4, and T^d1 Ad3, where AJ = k J d', and mJ d2Ad4 = O. From
Eqs. (4.2) and (8.51) we find the projection operator
= U2 —d.’ Ad3 ® (e! A e3 —a A e4) — d2 A d4 ® e2 А (m/e24-e4),
(8.52)
where a = C| Ac3 L m. Substituting Eqs. (8.51) and (8.52) into Eq. (4.6), one can find also the
differential characteristic operator of the layer.
If m takes one of the following values:55’56
m = a1(e3)1/2e1+a3(e1)l/2c3,
(8.53a)
where
m-=a|€2(e3) |/2С1 + <х3е2(в|) 1/2c3,
(8.53b)
(8.54)
and the signs of a । and a3 may be chosen arbitrarily, the operator has the only eigenvalue
£]— 0, in other words, JT’ becomes a nilpotent operator. The first type of degeneracy [with m
(8.53a)] has been investigated in Refs. 55, 56. Consider here the second type.
From Eqs. (4.6), (4.8), (8.51), (8.52), and (8.53b) one finds
JT'==A:4[(^1 AO34-<7 Ad4/e2) ®e2Aa + d2 A (m ® т-4-е — e2]3--d4® em) Ae4],
(8.55a)
t7X2-(A'4)25[(/?7/62-d4) Ad2® c2Aa+(d1 Ad3T^Ad4/e2) ®с4Ат], (8.55b)
Ж3 = (/с4) 362 (m/б2 - d4) A d2 ® c4 A m, (8.55c)
(8.55d)
JT4 = 0,
exp(/x J r),
(8.56)
where <52—(^ — e2) (б2 — £з)' Hence, the evolution operator У (8.56) describes the wave F
(4.11) with cubic amplitude dependence on f=xJQ=x2. The reflection and transmission
operators r, R and d, D of the layer are defined by the general expressions (7.7)—(7.10) and
(7.12), where Q-e2, M2=M (8.51), ,Т2=У~ (8.52), ^2-Ж (8.55a) and
X (М3-ЛО]схр(/(Х1-х2) J r).
(8 57)
The field values in the layer depend on the amplitude F(x0 of the incident wave as follows:
F2(x) = F(x—xl)F2(xl)t where F2(xj) = S'2j(U2+F)F(xI) is the boundary value of the
field, f1<xJ^2<f2> and ^2l, R and // are defined by Eqs. (4.15), (7.12), and (8.56). If
F2(x,) satisfies one of the following conditions:
m A e4 J F-(x, )^0;
(8.58a)
m A e4 J F~(Xl) =0,
е2Л a J F2(x1)^0;
m A e4 J F2 (x j) = 0,
е2Ла J F?(x]) =0,
.5TF2(X1)#:O;
(8.58b)
(8.58c)
the dependence of F2(x) on x2 is cubic, quadratic, and linear, respectively, At last, if F^'F2(x1)
—О, F2(x) reduces to the eigenwave F2(x) =F2(x1)exp[/(x — x1) J т].
C. Material parameters of dispersive linear media
In a dispersive linear medium, a value G(x) of the two-form G at point x depends on values
of the two-form Fin the vicinity of this point, i.e., in a homogeneous dispersive medium, G and'
Fare related by the integral transformation
^(y)F(x~y)<74^,
(8.59)
where d^y — dyx dy2 dy2 dy* is the infinitesimal element of the space-time volume, and is
some tensor function (^/ZeJ22). Let J/ be the evolution operator of some wave propagating in
such a medium, i.e., at any values of x and y, we have F(x + y)=J> (y)F(x). Then, Eq. (8.59)
reduces to the algebraic relation G(x) =M(Fs )F(x), where
//(У)-^(”У)А.
(8.60)
ГЬе material tensor )g/122 is defined on a set of evolution operators FF for which
there exists the integral in Eq. (8.b0). A similar approach was suggested in Ref. 56, where, on
the basis of the three-dimensional description in the rest frame of a medium, the generalized
tensors of permittivity б( у ), permeability ju(JF), and gyrotropy a (J* ) and /3(-л ) were
introduced. For plane waves in an anisotropic medium with the constitutive relations D —eE,
H = B, this approach was developed in detail in Ref. 54. It allows one to use complete sets of
plane-wave solutions of Maxweirs equations, including the Voigt waves, and has also some
other merits.
In this article, we consider electromagnetic waves with the evolution operator (4.9b).
For such waves, Л/(.А ) (8.60) can be expressed through Fourier components
?>/(/<) -
..// (у)exp(— iy J k)cfy
(8.61)
as follows
where idIf 3F has a simple structure, this formula simplifies:
;-i,...,7V.
Thus, in a uniformly moving dispersive linear medium, a wave with the evolution operator
У (4.9b) [see also Eq. (4.8)] is described by the Eqs. (3.4), (3.5), (4.1) and so on, where А/
depends on .7^ and t. The theory of wave propagation in dispersive media, based on the use of
the generalized material tensor 4f(/z ) (8.62), will be presented separately.
IX. INVERSE PROBLEMS
The developed above technique makes it possible to find the solution of the inverse problem
of reflection and transmission for an anisotropic and gyrotropic medium at a few very general
propositions about the properties of the medium such as its homogeneity and linearity.
A. Riccati’s equation for the surface impedance operator
From Eqs. (3.4), (3.5), and (6.14)—(6.18) we obtain
/4-Г=(у4-/?;)(Л')-(у-^)4-С, (9.1)
— cp = iN'(p, (9.2a)
= (9.2b)
where
А'=А—Ат® тА/Л, (9.3a)
B'=B\ + B2, (9.3b)
B'l = Bl + Blr^QB]/^, (9.3c)
B2 = B2+B2Q®tB2/A, (9.3d)
C' = C-CQ®QC7A, (9.3e)
A = (QA-t)M(QAt), (9.3f)
and A, Blt B2, C are defined by Eq. (5.7), A', B\, B2, C as well as у satisfy the relation
(6.18a), and IN' =N'I=N'. For the wave f (6.2), у and N' are independent on f. Hence,
q?(£) =cxp(f£7V')<p(0), and Riccati’s equation (9.1) becomes
(y+5J)(J')-(y-^)+C'=0. (9.4)
Let y+ and y_ be two different solutions of Eq. (9.4), and N’* = (zl')-(y± — Then,
from Eqs. (9.2b)-(9.4), wc obtain
T = (y+—y_)x ,
C'=(y+-y_)2V'_x-7V;,
^ = (y^V'_-y+A^)x-,
2?2 = y_^_7V'+ — y+x’WL
(9.5a)
(9.5b)
(9.5c)
(9.5d)
where x = — N'__, and и = 2(xtI—x')/(x/\ Ax), is a pseudoinverse operator (x x
— xx-—Z, Ix~ — и I = %~ ).
B. Calculation of and N\
Consider a layer of the medium under investigation surrounded by media with known
material tensors Mx and M2. Let yA+, Nk+ and yk_, Nk_ be the operator parameters of
(total) waves (see Secs. VI and IX A) propagating in the Zth medium (/с—1,2,3) in the
positive and negative C directions, respectively. For convenience, we shall omit the subscript 2
attached to each parameter related with the second medium, i.e., M = M2,
N ~N'2 , and so on. Given Q, r, , and Af3, the operators jq ± and y3± can be found from
Eq. (6.20) or, if the layer is surrounded by an isotropic media or vacuum, from Eqs. (8.25),
(8.27), or (8.29).
For the incident wave propagating in the first medium, the reflection operator rx of the first
boundary and the transmission operator d} of the layer are given by (see Secs. VII A and
IX A)
П = (Г+-Г1-) (Г1 + -7+),
(9.6a)
</| = (П+-7-) (Г|.-Г_)ехр(/£7У'ь)(у+-У|_)' (yi + -y,_),
(9.6b)
where f=(x2 —xj) J Q, and the points on the boundaries are chosen so that (x2 —xj J t = 0.
We assume here that the layer is sufficiently thick so that the reflected and transmitted waves
under study can be separated from multiple reflected beams.
Using /q, (9.6) and the similar operators r3, dy for the wave incident from the third
medium, we obtain
у+ = (/1 ++У1_Г1)(7+г1) ,
dd,
/V; = -/(7 + ^)(</,)- -47
(7+f|) ,
у_ = (П-+Г1 + ''з)(7+г3) -
dd}
N’_=i(J+r}Hd}'T (7+r3)-.
(9.7a)
(9.7b)
(9.7c)
(9.7d)
Thus, to find y , , 2V'( and then A', B\, B2, C (9.5), it is necessary to measure the
reflection and transmission operators г,, r3 and d\, d2. On a similar basis, a method of dielectric
permittivity measurement for a motionless nongyrotropic nonmagnetic anisotropic medium
was suggested in Ref. 66.
The operators r,, dx, ry, and having been measured, A', B\, B'2, and C having been
determined then from Eqs. (9.5) and (9.7), one can find the material tensor M as follows.
C. rzdculation of the material tensor M
1 et (b,) and (/>') be dual bases, i.e , b, J fJ = Sf, i,k— 1,2,3,4, and
b! = n L (^Afl’A/T*)//,,, (9.8a)
Ь^-ПК/З'Л/З’А/З4)/^,
(9.8b)
b3=nL(j31A^2Aj94)/A0>
b4=-ll L (Д'ЛД2 A £’)//„,
where Ло= —fl L (/З1 Л/32 A/3’ Л/34). The material tensor M can be written as
E Л0'ЛЬЧЛ1>/,
1 ч/<y<4
I<A<Z<4
(9.8c)
(9.8d)
(9.9)
where M\k> — (P, А A ft). To find all 36 components М\к1, it is necessary to carry out
the measurements at different values of Q and t. In particular, one can use a specimen having
the form of parallelepiped with boundaries defined by some one-forms Q', /’=1,2,3. In this case,
it is convenient to select the basis (/3') as follows: /3'= Q', z = 1,2,3, ft f\ft f\ft /\ft^0. Let us
assume also that the measurements are carried out for each boundary at three different values
of r, namely, t1, t2, and t3, such that r1 Ar2 A ft AQ'-A-O, Q1 AQ2AQ3At5A0, i—1,2,3.
The relations (9.3a) and (9.3c) can also be written as
7/ГГ=(Л')Л
ic~r = (C')~.
(9.10a)
(9.10b)
Using the dual bases (ny) and (v7), where v7 = r7, j = 1,2,3, v4 = O', from Eqs. (9.5a) and
(9.10a) we obtain
4=QrJMLQ'’
— _|_£-(g2) ® T|— (п2£'(',|)л2)т2 ® т2 — (щЕ^п3 )ft ® r3] ,
(9.11)
£(/./•) = (N’+ _N’_) (y+ _r_) - | Q=Q, т=тЛц=П4 v=n^. (9.12)
Here, (n,) and (vf) arc related by Eq. (9.8), and the pseudoinverse tensor in Eq. (9.11) is
calculated by the formula (Al) in which u and и arc replaced by Q; and Q'. The relations
(9.7), (9.11), and (9.12) express directly At (z=i,2,3) by the reflection and transmission
operators. On other hand, from Eq. (9.9) we have
4
At = X (9.13a)
j.l=2
A2 = (M[22bl-M'23l2b3-M'2412b4) ® b,+ (-M'n23b' + M'2323b3+M’2423b4) ® b3
+ (-M\224b4M2i24b3+M'24ub4)®b4, (9.13b)
а3^(м\313ь[+м23'3ь2-м34'3ь4) ®ь1 + (л/;321//+л/3323л2-л/;,12^4) «ь2
+ {-M\334b3-M'2334b2 + M3434b4) ®b4. (9.13c)
Since AfJ,u = PjAjP1, M232 = — P^A^P3, and so on, the obtained above relations make it
possible to find 24 of 36 material parameters М'^к.
To find the other twelve components of M, let us first express Cj = Tj J M L? (j = 1,2,3)
by 7V'± and . Using the relations (9.5b), (9.10b), and the dual bases (m,) and («'), where
&)'=Q', z=l,2,3, w4=t7 and («') defines (m,) by Eq. (9.8), we obtain
С} = Tj J M L tj
= [^(1'7)+№2’7)+A'l3'7)-(ni1^(3'7’/n1)Ql®Q1
-(m2X(,'7)m2)Q2 ®Q2-(m3A'(2'7>Z7i3)Q3 ®Q3]-, (9-14)
'•»=[(?/'_)--(JV'+)-](r+-y_)-|Q = Q'. r=r>. u = m;., v = m4- (9-15)
The pseudoinverse tensor in Eq. (9.14) is calculated by the formula (Al) in which u and и are
replaced by г,- and r7. To find M, it is convenient to set (r7) as follows:
+ (9.16a)
t2 = ^/?,+t^3 + t^4, (9.16b)
т3 = т^1 + т^2 + т3/У4. (9.16c)
Then, from Eqs. (9.9) and (9.14), we find
m;424=(t3)-2(q1c3q2+t]t3m;2,2+73t3m;4,2-t3t3m;224), (9.i?a)
м^|4=(73)-2(о2с3а,+7373м;212+^3м;214-737М12). (9.17b)
m;434 = (t2 ) -2 ( q , c2q3+t7t2m;3 13+t2t2m;413 - Ht2m;334) , (9. nC)
m;414=(t2)-2(q3c2q,+t2t2m;3I3+t272m;3,4-t^2m;413), (9.i7d)
M ;434 = (r\) -2 (Q2G Q3 + + т’r>;423 - т' t\M2334), (9.17e)
m;424=(4) 2«2зС1а2+7;7>2323+т;7>ь24-^Жм2’)- (9.17D
By preliminary computation of B\ and Бэ, from Eqs. (9.3), (9.5c), (9.5d), and (9.16) we
obtain the last six components
;W;234=[riM;223 + Q|^(2'1)Q3-(Ql/l2Tl)(Q2C1Q3)/Z21]/7t , (9.18a)
M;412=[rbV3124-Q3^(2J)Q1-(r1^2Q1)(Q3C1Q2)/Z21]/T', (9.18b)
xV'314=L-7->;313 + Q27?;IU)Q1-(Q2/13t2)(Q3C2Q1)/A32]/t2, (9.18c)
M ;423 = [ - + Q ] B\<3’2 ’ Q2 - (T2/l 3Q2) (Q) C2Q3) /Л32 ] /т2, (9.18d)
v;,24 |Г|Л/:,12 Q(«;(,'"Q2 I-(Q3/11t3)(QiC3Q2)/Ti3|/t3, (9.18cj
л/'4'’=[т>/-;2|3-с2/?;(1-,'о,-|-(тгЛ,а3)(о2с,о')/л|3]/т3, (9.i8f)
’..'here /.jj — TjAjT2, and
B{^ = (r_NL-r+N'+)(N'+-N'_')~\Q^, T~Tj, (9.19a)
B’2(i'» = [yAN'+-N'_)-N'+-y+{N\ -7V'_)-^_] |q=q-. t=tJ. (9.19b)
All components M'.^ on the right-hand side of Eqs. (9.17) and (9.18) have been found above
at the first stage of calculation [see Eqs. (9.11)—(9.13)].
On the whole, the measurement of the material tensor M consists of the following three
stages: (1) the measurement of the reflection and transmission operators r1} and r/15 at
different values of Q and t; (2) the calculation of the surface impedance operators and the
operators by the formulas (9.7); (3) the calculation of the material parameters М‘к1Ъу the
formulas (9.11 )-(9.19).
We obtained above the solution of the inverse problem by rather general assumptions about
the properties of the medium and its boundaries. In particular, since all obtained above rela-
tions are formulated in the Lorentz covariant form, the solution holds true when the medium
and its boundaries are moving with different velocities with respect to the laboratory frame.
X. CONCLUSION
The suggested intrinsic tensor technique considerably facilitates the solution of various
boundary value problems, including inverse problems, in the electrodynamics of uniformly
moving media. Besides the operations of exterior algebra with its standard system of notation,
this formalism allows one also to use a wide spectrum of operations defined for antisymmetric
tensors which describe linear mappings in the spaces of /-vectors and л-forrns. This formalism
constitutes a natural mathematical basis for the generalization of the impedance method and
the characteristic matrix method on the case of moving media. The advantages of the developed
techniques are illustrated by applying them to some problems in relativistic electrodynamics.
ACKNOWLEDGMENT
I am grateful to the referee for constructive comments and for valuable suggestions.
APPENDIX: SOME TYPES OF PSEUDOINVERSE TENSORS
In this article, we use several types of pseudoinverse tensors. Consider first a tensor AeT/
satisfying the conditions /1Q = O, Q/1—0, А А AAA AA^AO. In this case, the pseudoinversc
tensor
1
/1“-— (*w)l (ЛЛЛЛ)1 (*u) (Al)
ZO
has the properties A~A — Uj — Q <8 u, AA “--Uj —w®Q, where uJQ— 1, and 6= (1/6)
X(*u)J (A A AAA A A) L (*u). The proof immediately follows from the dyad expansion of
И.
Consider now the tensors
% = ЦМ12 — — и Л А Л u.
к= - (1/2)П1 (A A A/l)l П,
(A2a)
(A2b)
where /Ь-QJMLQ. It follows from Eqs. (4.3), (Al), and (A2) that
(A3a)
хх~ВЦ>
(A3b)
= ВЦ.
(A3c)
If 5^0, then x and x define the linear operators of rank 3 in Л2( V) and Л2( И*), and x — x/B
is the pseudoinverse tensor. If the operator A has a two-dimensional kernal (6 — 0,
А Л ЛЛ7МЗ), in other words, if A can be written as a sum of two dyads, then x (A3a) takes the
form x — where асЛ2(И), rue Л2(К*), ax —0, xco — 0.
The other two types of pseudoinverse tensors are described in Secs. VII A and IX A.
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